1 consumer preferences and the concept of utility chapter 3

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Consumer Preferences

and the Concept of

Utility

Chapter 3

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Chapter Three Overview

1. Motivation

2. Consumer Preferences and the Concept of Utility

3. The Utility Function• Marginal Utility and Diminishing Marginal Utility

4. Indifference Curves

5. The Marginal Rate of Substitution

6. Some Special Functional Forms

1. Motivation

2. Consumer Preferences and the Concept of Utility

3. The Utility Function• Marginal Utility and Diminishing Marginal Utility

4. Indifference Curves

5. The Marginal Rate of Substitution

6. Some Special Functional Forms

Chapter Three

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Motivation

• Why study consumer choice?

• Study of how consumers with limited resources choose goods and services

• Helps derive the demand curve for any good or service

• Businesses care about consumer demand curves

• Government can use this to determine how to help and whom to help buy certain goods and services

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Consumer Preferences

Consumer Preferences tell us how the consumer would rank (that is, compare the desirability of) any two combinations or allotments of goods, assuming these allotments were available to the consumer at no cost.

These allotments of goods are referred to as baskets or bundles. These baskets are assumed to be available for consumption at a particular time, place and under particular physical circumstances.

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Consumer Preferences

Preferences are complete if the consumer can rank any two baskets of goods (A preferred to B; B preferred to A; or indifferent between A and B)

Preferences are transitive if a consumer who prefers basket A to basket B, and basket B to basket C also prefers basket A to basket C

A B; B C = > A C

A B; B C = > A C

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Consumer Preferences

Preferences are monotonic if a basket with more of at least one good and no less of any good is preferred to the original basket.

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Types of Ranking

Students take an exam. After the exam, the students are ranked according to their performance. An ordinal ranking lists the students in order of their performance (i.e., Harry did best, Joe did second best, Betty did third best, and so on). A cardinal ranking gives the mark of the exam, based on an absolute marking standard (i.e., Harry got 80, Joe got 75, Betty got 74 and so on). Alternatively, if the exam were graded on a curve, the marks would be an ordinal ranking.

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The Utility Function

Chapter Three

The three assumptions about preferences allow us to represent preferences with a utility function.

Utility function

– a function that measures the level of satisfaction a consumer receives from any basket of goods and services.

– assigns a number to each basket so that more preferred baskets get a higher number than less preferred baskets.

– U = u(y)

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The Utility Function

• An ordinal concept: the precise magnitude of the number that the function assigns has no significance.

• Utility not comparable across individuals.

• Any transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences. e.g. U = vs. U = + 2 represent the same preferences.

Chapter Three

y y

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Marginal Utility

Marginal Utility of a good y

• additional utility that the consumer gets from consuming a little more of y

• i.e. the rate at which total utility changes as the level of consumption of good y rises

• MUy = U/y

• slope of the utility function with respect to y

Marginal Utility of a good y

• additional utility that the consumer gets from consuming a little more of y

• i.e. the rate at which total utility changes as the level of consumption of good y rises

• MUy = U/y

• slope of the utility function with respect to yChapter Three

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Diminishing Marginal Utility

The principle of diminishing marginal utility states that the marginal utility falls as the consumer consumes more of a good.

The principle of diminishing marginal utility states that the marginal utility falls as the consumer consumes more of a good.

Chapter Three

Diminishing Marginal Utility

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Marginal Utility

The marginal utility of a good, x, is the additional utility that the consumer gets from consuming a little more of x when the consumption of all the other goods in the consumer’s basket remain constant.

•U(x, y) = x + y U/x (y held constant) = MUx

• U/y (x held constant) = MUy

The marginal utility of a good, x, is the additional utility that the consumer gets from consuming a little more of x when the consumption of all the other goods in the consumer’s basket remain constant.

•U(x, y) = x + y U/x (y held constant) = MUx

• U/y (x held constant) = MUy

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Marginal Utility

Example of U(H) and MUH

U(H) = 10H – H2

MUH = 10 – 2H

Example of U(H) and MUH

U(H) = 10H – H2

MUH = 10 – 2H

Chapter Three

H H2 U(H) MUH

2 4 16 64 16 24 26 36 24 -28 64 16 -6

10 100 0 -10

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Marginal Utility

Chapter Three

U(H) = 10H – H2

MUH = 10 – 2H

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Marginal Utility

Example of U(H) and MUHExample of U(H) and MUH

Chapter Three

• The point at which he should stop consuming hotdogs is the point at which MUH = 0

• This gives H = 5. • That is the point where Total Utility is flat.• You can see that the utility is diminishing.

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Marginal Utility – multiple goodsU = xy2

MUx = y2

MUy = 2xy

U = xy2

MUx = y2

MUy = 2xy

Chapter Three

• More is better? More y more and more x indicates more U so yes it is monotonic• Diminishing marginal utility? • MU of x is not dependent of x. So the marginal utility of x (movies) does not decrease as the number of movies increases.• MU of y increases with increase in number of operas (y) so neither exhibits diminishing returns.

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Indifference Curves

An Indifference Curve or Indifference Set: is the set of all baskets for which the consumer is indifferent

An Indifference Map : Illustrates a set of indifference curves for a consumer

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Indifference Curves

1) Monotonicity => indifference curves have negative slope – and indifference curves are not “thick”

2) Transitivity => indifference curves do not cross

3) Completeness => each basket lies on only one indifference curve

1) Monotonicity => indifference curves have negative slope – and indifference curves are not “thick”

2) Transitivity => indifference curves do not cross

3) Completeness => each basket lies on only one indifference curve

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Indifference Curves

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Indifference Curves

Suppose that B preferred to A.but..by definition of IC,B indifferent to CA indifferent to C => B indifferentto C by transitivity.

And thus a contradiction.

Suppose that B preferred to A.but..by definition of IC,B indifferent to CA indifferent to C => B indifferentto C by transitivity.

And thus a contradiction.

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Indifference Curves

U = xy2

Check that underlying preferences are complete, transitive, and monotonic.

Chapter Three

2yMU x

xyMU y 2x y xy^2

8 4.24 143.84 6 1443 6.93144.071 12 144

144Ufor

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Indifference Curves

Example: Utility and the single indifference curve.

Example: Utility and the single indifference curve.

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Marginal Rate of Substitution

The marginal rate of substitution: is the maximum rate at which the consumer would be willing to substitute a little more of good x for a little less of good y;

It is the increase in good x that the consumer would require in exchange for a small decrease in good y in order to leave the consumer just indifferent between consuming the old basket or the new basket;

It is the rate of exchange between goods x and y that does not affect the consumer’s welfare;

It is the negative of the slope of the indifference curve:

MRSx,y = -y/x (for a constant level of preference)

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If the more of good x you have, the more you are willing to give up to get a little of good y or the indifference curves get flatter as we move out along the horizontal axis and steeper as we move up along the vertical axis

If the more of good x you have, the more you are willing to give up to get a little of good y or the indifference curves get flatter as we move out along the horizontal axis and steeper as we move up along the vertical axis

Chapter Three

Marginal Rate of Substitution

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-y/x =

MUx(x) + MUy(y) = 0 …along an IC…

MUx/MUy = MRSx,y

Positive marginal utility implies the indifference curve has a negative slope (implies monotonicity)

Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

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Marginal Rate of Substitution

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Implications of this substitution:

• Indifference curves are negatively-sloped, bowed out from the origin, preference direction is up and right • Indifference curves do not intersect the axes

Implications of this substitution:

• Indifference curves are negatively-sloped, bowed out from the origin, preference direction is up and right • Indifference curves do not intersect the axes

Chapter Three

Marginal Rate of Substitution

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Indifference Curves

Chapter Three

Averages preferred to extremes => indifference curves are bowed toward the origin (convex to the origin).

29Chapter Three

Indifference Curves

Do the indifference curves intersect the axes?

A value of x = 0 or y = 0 is inconsistent with any positive level of utility.

Do the indifference curves intersect the axes?

A value of x = 0 or y = 0 is inconsistent with any positive level of utility.

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Marginal utilities are positive (for positive x and y)

Example: U = Ax2+By2; MUx=2Ax; MUy=2By

(where: A and B positive)

MRSx,y = MUx/MUy = 2Ax/2By = Ax/By

Marginal utility of x increases in x;

Marginal utility of y increases in y

Chapter Three

Marginal Rate of Substitution

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Example: U= (xy).5;MUx=y.5/2x.5; MUy=x.5/2y.5

A. Is more better for both goods? Yes, since marginal utilities are positive for both. B. Are the marginal utility for x and y diminishing? Yes. (For example, as x increases,for y constant, MUx falls.)

C. What is the marginal rate of substitution of x for y?

MRSx,y = MUx/MUy = y/x

Chapter Three

Indifference Curves

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Example: Graphing Indifference Curves

IC1

IC2

x

y

Preference direction

Chapter Three

Indifference Curves

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Cobb-Douglas: U = Axy

where: + = 1; A, , positive constants

MUX = Ax-

1y MUY = Ax y-1

MRSx,y = (y)/(x)

“Standard” case

Chapter Three

Special Functional Forms

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Example: Cobb-Douglas (speed vs. maneuverability)

IC1

IC2

x

y

Preference Direction

Chapter Three

Special Functional Forms

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Perfect Substitutes: U = Ax + By

Where: A, B positive constants MUx = A MUy = B MRSx,y = A/B so that 1 unit of x is equal to B/A units of y everywhere (constant MRS).

Where: A, B positive constants MUx = A MUy = B MRSx,y = A/B so that 1 unit of x is equal to B/A units of y everywhere (constant MRS).

Chapter Three

Special Functional Forms

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Example: Perfect Substitutes • (Tylenol, Extra-Strength Tylenol)

x0

y

IC1IC2 IC3

Slope = -A/B

Chapter Three

Special Functional Forms

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Perfect Complements: U = Amin(x,y) where: A is a positive constant. MUx = 0 or A MUy = 0 or A MRSx,y is 0 or infinite or undefined (corner)

Perfect Complements: U = Amin(x,y) where: A is a positive constant. MUx = 0 or A MUy = 0 or A MRSx,y is 0 or infinite or undefined (corner)

Chapter Three

Special Functional Forms

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Example: Perfect Complements • (nuts and bolts)

x0

y

IC1

IC2

Chapter Three

Special Functional Forms

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U = v(x) + Ay

Where: A is a positive constant.

MUx = v’(x) = V(x)/x, where small MUy = A

"The only thing that determines your personal trade-off between x and y is how much x you already have." *can be used to "add up" utilities across individuals*

"The only thing that determines your personal trade-off between x and y is how much x you already have." *can be used to "add up" utilities across individuals*

Chapter Three

Special Functional Forms

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Example: Quasi-linear Preferences • (consumption of beverages)

••

IC’s have same slopes on anyvertical line

x

y

0

IC2

IC1

Chapter Three

Special Functional Forms